Firing an employee for complaining about their wages will violated two laws.
Employees have probably complained about wages since there have been wages. As a result laws have been created that protect the right to complain. Today any employer who wants to fire an employee for complaining about their pay does so at their own peril.
Two laws
There are two laws that protect the rights of employees to complain about their wages. The first of these has received a great deal of attention of late. The National Labor Relations Act, Section 7 gives covered employees the right to engage in the “protected concerted activity” of discussing wages, hours and working conditions. As long as two or more employees wish to discuss wages for the purpose of improving wages they are protected from doing so. Firing those employees results in a violation of the NLRA that may result in reinstatement and back pay for the fired employees and may get the company some unwanted attention from a union.
However, not all employees are covered by the NLRA. Management employees, supervisors and HR employees are not covered by this act and thus don’t have that protection. However, they do have some protection under the Fair Labor Standards Act.
Retaliation provision
Under the FLSA there is a provision that protects employees who have been terminated for complaining about their pay. It has always been the case that if an employee went to the Wage and Hour division of the USDOL to file a complaint and were subsequently fired the company could be charged with retaliation and sued. Now however, the Second Circuit of the US Court of Appeals has ruled that a complaint does not have to be filed with a government agency to be protected. The act of simply complaining to an employer is enough to invoke the protection against retaliation in the FLSA. This puts the Second Circuit in agreement with the First and Ninth Circuits, thus making it likely that this stance will become much more widespread.
Just bad HR
In my opinion firing employees for complaining about their pay is just bad HR to begin with. If employees have those complaints it is something that needs to be listened to and dealt with in order to improve relations with those employees. If you have just a single employee who is a chronic complainer there will different opportunities to deal with the employee other than firing them over wage complaints.
The wisest course of action is to recognize that you have a problem, address the problem and try to determine a solution that will make everyone happy. Retaliation just leads to lawsuits.